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Full text of "Mathematical And Physical Papers - Iii"

ON THE MOTION OF PENDULUMS.                          23
dynamics, will not be without its influence when friction is taken into account; but the effect is so very small in practical cases that it is not worth while to take it into account. For if a be the radius of the sphere, and I the length of the suspending wire, the velocity of a point in the surface of the sphere due to the motion of rotation will be a small quantity of the order a/I compared with the velocity due to the motion of translation. In finding the moment of the pressures of the fluid on the pendulum, forces arising from these velocities, and comparable with them, have to be multiplied by lines which are comparable with a, I, respectively. Hence the moment of the pressures due to the motion of rotation of the sphere will be a small quantity of the order a2/?, compared with the moment due to the motion of translation. Now in practice I is usually at least twenty or thirty times greater than a, and the whole effect to be investigated is very small, so that it would be quite useless to take account of the motion of rotation of the sphere.
The problem, then, reduces itself to this.    The centre of a sphere performs small periodic oscillations along a right line, the, sphere itself having a motion of translation simply: it is required to determine the motion of the surrounding fluid.
10. Let the mean position of the centre of the sphere be taken for origin, and the direction of its motion for the axis of x, so that the motion of the fluid is symmetrical with respect to this axis. Let or be the perpendicular let fall from any point on the axis of x, q the velocity in the direction of TV, co the angle between the line us and the plane of xy. Then p, u} and q will be functions of x, -GT, and t, and we shall have
v = q cos a), w = q sin co, y  w cos co, z = w sin CD, whence                     -or2 = y* 4- 02,    co = tan"1 - .
We have now to substitute in equations (2) and (3), and we are at liberty to put co = 0 after differentiation. We get
d                 d      sin co  d         d     ,              ~
-y- = cos ft) -j----------------- > = ~T~ when a) = 0,
dy              d'sr       '&   do)       ut*T
= -== when co = 0. d\f    d&